Biostatistics

698 CHAPTER 13 NONPARAMETRIC AND DISTRIBUTION-FREE STATISTICS
13.7 THE KOLMOGOROV–SMIRNOV
GOODNESS-OF-FIT TEST
When one wishes to know how well the distribution of sample data conforms to some
theoretical distribution, a test known as the Kolmogorov–Smirnov goodness-of-fit test
provides an alternative to the chi-square goodness-of-fit test discussed in Chapter 12. The
test gets its name from A. Kolmogorov and N. V. Smirnov, two Russian mathematicians
who introduced two closely related tests in the 1930s.
Kolmogorov’s work (6) is concerned with the one-sample case as discussed here.
Smirnov’s work (7) deals with the case involving two samples in which interest centers on
testing the hypothesis that the distributions of the two-parent populations are identical. The
test for the first situation is frequently referred to as the Kolmogorov–Smirnov one-sample
test. The test for the two-sample case, commonly referred to as the Kolmogorov–Smirnov
two-sample test, will not be discussed here.
The Test Statistic In using the Kolmogorov–Smirnov goodness-of-fit test, a
comparison is made between some theoretical cumulative distribution function, F T (x), and
a sample cumulative distribution function, F S (x). The sample is a random sample from a
population with unknown cumulative distribution function F(x). It will be recalled (Section
4.2) that a cumulative distribution function gives the probability that X is equal to or less
than a particular value, x. That is, by means of the sample cumulative distribution function,
F S (x), we may estimate PX ð xÞ. If there is close agreement between the theoretical and
sample cumulative distributions, the hypothesis that the sample was drawn from the
population with the specified cumulative distribution function, F T (x), is supported. If,
however, there is a discrepancy between the theoretical and observed cumulative distribution
functions too great to be attributed to chance alone, when H 0 is true, the hypothesis
is rejected.
The difference between the theoretical cumulative distribution function, F T (x), and
the sample cumulative distribution function, F S (x), is measured by the statistic D, which is
the greatest vertical distance between F S (x) and F T (x). When a two-sided test is appropriate,
that is, when the hypotheses are
H 0 : Fx ðÞ¼F T ðÞ x for all x from 1 to þ1
H A : Fx ðÞ6¼ F T ðÞ x
for at least one x
the test statistic is
D ¼ sup
x
jF S ðÞ x F T ðÞ x j (13.7.1)
which is read, “D equals the supremum (greatest), over all x, of the absolute value of the
difference F S ðXÞ minus F T ðXÞ.”
The null hypothesis is rejected at the a level of significance if the computed value
of D exceeds the value shown in Appendix Table M for 1 a (two-sided) and the sample
size n.